Quick way to clear Calendar notifications

Actually, this functionality has been moved into the Notification settings for Calendar items (Settings > Notifications > Calendar).

In that view you'll have the option to change notifications for: Upcoming Events; Invitations; Invitee Responses; and Shared Calendar Changes. Each one give the ability to select: whether its shown in notification center; notification sound; badge app icon; show on lock screen; and the alert style!

You cannot silence the alerts of a single calendar in iOS. Alerts utilize Notification Center, and the Calendar options in Notification Center apply to all calendars. I know this is not the answer you are hoping for.

The best option, as mentioned in the comments above, is to disable the calendar on your work account, when not desired, and re-enable it again when you need it. To do this, go to Settings > Mail, Contacts, Calendars > Your Work Mail Account. Slide off Calendars. This isn't a good option, admittedly, as you will have to re-download your calendars every time it's slid back on.

Another option is to keep calendars turned off for this account, and use Exchange Outlook Web Access to access your calendars. This may be even a worse interface, however, as the OWA interface is not mobile-optimized.

My last suggestion is to use a third-party app to manage one of your calendars and the built-in Calendar app to manage the other. You can then disable Notifications on one of the two apps. You have to look potentially in two places for your calendar entries, but you will potentially get the granular control you need. Here are some calendar apps to examine: Calvetica, MiCal.